This was primarily due to the 1557 influenza pandemic, which returned in 1558 and perhaps lingered for another year or two. This was a global pandemic and other areas of Europe were also severely hit. Making things even worse was that the influenza was preceded by plague, typhus, measles (hat tip: Rusl) and famine in some regions of Europe.
Influenza significantly contributed to England's unusually high death
rates for 1557–58: Data compiled on over 100 parishes in
England found that the mortality rates increased by up to 60% in some
areas during the flu epidemic...
As a result, the population of England fell by an estimated 2% between 1557 and 1559 and deaths for the year 1558 were 80% above normal.
This pandemic was a global one and
was highly fatal, with deaths recorded as being due to pleurisy and
fatal peripneumony. It first infested Asia, then Constantinople, and
having spread all over Europe, afterwards attacked America. Before
autumn 1557, it simultaneously hit all parts of Spain so quickly that:
«the greater part of the population in that Kingdom were seized with
it almost on the same day»
Thomas Short described the epidemic based on contemporary reports, the
disease: «came from the land Melite in Africa, into Sicily; so into
Spain, and Italy […]. It attacked at once, and raged all over Europe,
not missing a family and scarce a person.
Source: Rosamaria Alibrandi, 'When early modern Europe caught the flu. A scientific account of pandemic influenza in sixteenth century Sicily'. In Medicina Historica 2018; Vol. 2, N. 1: 19-26.
A Note on Life Expectancy
Note that the life expectancy figures quickly rose again, and that the source of the data in the question states that:
Shown is period life expectancy at birth, the average number of years
a newborn would live if the pattern of mortality in the given year
were to stay the same throughout its life.
or, to put it in a slightly different way,
The statistic “Life expectancy at birth” actually refers to the
average number of years a newborn is expected to live if mortality
patterns at the time of its birth remain constant in the future. In
other words, it’s looking at the number of people of different ages
dying that year, and provides a snapshot of these overall “mortality
characteristics” that year for the population.
Some Historical Context
Also worth noting is that the 1557 pandemic was not the first, but it is better documented than the ones that came before it:
...although influenza has been known in Europe since the middle of the
13th century and there are some records documenting six visitations in
the 14th century, and four in the 15th, it only began to be studied by
the profession from the start of the following century. Only then
records of erupted epidemics appeared, together with the circumstances
which attended their outbreak and progression, and their